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Investing in commodities Will commodity futures enhance risk-‐adjusted return in efficient portfolios?
Bachelor thesis, Financial Economics, 15 HP
Department of Economics
Authors:
Eric Öström 890804-‐4970
Per Svennerholm 890313-‐5096
Supervised by:
Charles Nadeau, School of Business Economics and Law
Abstract
With this paper we intend to investigate what kind of benefits there are by adding
commodity futures to a well-‐diversified portfolio. Since the last fifteen years the
commodity speculation has grown tremendously, which partially can be explained by
that commodities exposes the investor to certain factors other than an investment in
equities. According to our calculations the commodity futures have outperformed stocks
during our research period, which partially could be explained by the increasing
demand of physical commodities in developing countries e.g. India & China (Akey,
2005). By constructing different portfolios consisting of equities and corporate bonds
we could investigate whether our portfolios will benefit from commodity futures and
how this will vary over different levels of risk. By using monthly data; 2002-‐2012, we
have concluded that by adding commodity futures to efficient portfolios, the return-‐to-‐
volatility increases. We have established that gold is a superior investment among the
commodity futures, during our sample period and geographical area. We have also
concluded that a Norwegian portfolio consisting of stocks and bonds benefit more, in
terms of return to volatility, than a Swedish portfolio by adding commodity futures.
3 Investing in Commodities, 2013
Table of Contents
1 Introduction .................................................................................................................................................. 5
1.1 Earlier empirical work ...................................................................................................................... 6
1.2 Hypotheses ............................................................................................................................................ 7
1.2.1 Hypothesis 1 ................................................................................................................................. 8
1.2.2 Hypothesis 2 ................................................................................................................................. 8
1.2.3 Hypothesis 3 ................................................................................................................................. 8
2 Theory .............................................................................................................................................................. 9
2.1 Modern Portfolio theory .................................................................................................................. 9
2.2 Sharpe Ratio ....................................................................................................................................... 10
2.3 Portfolio optimization .................................................................................................................... 10
2.3.1 Optimal risky portfolio .......................................................................................................... 10
2.3.2 Portfolio Variance .................................................................................................................... 11
2.4 Commodities ...................................................................................................................................... 11
2.5 Commodity investment vehicles ............................................................................................... 12
2.5.1 Exchange Traded Funds (ETF) .......................................................................................... 12
2.5.2 Mutual Funds ............................................................................................................................. 12
2.5.3 Equities in commodity based companies ...................................................................... 12
2.5.4 Future contract ......................................................................................................................... 13
2.6 Goldman Sachs Commodity Index (GSCI) .............................................................................. 13
2.7 Test for normally distributed returns ..................................................................................... 14
2.8 Criticism of MPT ............................................................................................................................... 15
3 Data ................................................................................................................................................................ 16
3.1 Stocks .................................................................................................................................................... 16
3.2 Commodities ...................................................................................................................................... 16
4 Investing in Commodities, 2013
3.3 Corporate Bond Funds ................................................................................................................... 17
3.4 Risk-‐free rate ..................................................................................................................................... 17
4 Methodology ............................................................................................................................................... 18
4.1 Bloomberg add-‐ins .......................................................................................................................... 18
4.2 Optimal Portfolio Allocation ....................................................................................................... 18
4.3 Test for normally distributed returns ..................................................................................... 19
5 Results ........................................................................................................................................................... 20
5.1 Swedish results ................................................................................................................................. 21
5.1.1 Asset Allocation ........................................................................................................................ 21
5.2 Norwegian results ........................................................................................................................... 23
5.2.1 Asset allocation ......................................................................................................................... 23
6 Analysis ........................................................................................................................................................ 26
7 Conclusions ................................................................................................................................................. 30
8 Methodology critique ............................................................................................................................. 32
9 Further research ....................................................................................................................................... 32
10 Bibliography ............................................................................................................................................ 33
11 Appendix ................................................................................................................................................... 37
11.1 Excel and VBA ................................................................................................................................. 39
11.2 Test statistics for Shapiro-‐Wilk test for normality ......................................................... 40
5 Investing in Commodities, 2013
1 Introduction
In the last years using commodity futures as an alternative investment in portfolios for
diversification purpose has become very popular. With new instruments and derivatives
using all kinds of commodities as underlying assets, it is nowadays as easy for an
individual speculating investor to trade commodity derivatives, as it is for an
institutional investment bank with decades of knowledge within the industry. The vast
majority of investors using commodity futures are institutional or commercial users that
are hedging against price movements in order to reduce financial loss. The other part of
investors using futures in their portfolios is most likely individuals who are speculating
in the price movements of the commodity (Investopedia, 2012). Speculating in price
movements will demand that the future contract is needed to be closed out before the
maturity date; otherwise the speculator might end up with 100 barrels of oil in physical
form.
However, one can argue that commodity futures can add diversification benefits to
portfolios because of many different reasons. With this paper we intend to investigate
these benefits from a perspective based on modern portfolio theory and to provide the
reader with empirical results based on historical data.
Stocks, mutual funds and bonds belong to the more traditional asset classes, while
commodity futures together with hedge funds and private equity form a more
alternative way to invest funds (UBS, 2011). Combining different asset classes give rise
to different covariance relationships. Why we found it interesting and important to see
how commodity futures affect efficient equity portfolios is because of the covariance
between commodity futures and stocks/bonds often tend to be low, and if this
relationship could be exploited to improve portfolio performance. Another aspect of the
importance is that commodity investing, has grown rapidly in volume the latest years
and our results might provide an explanation for this increase (CBOE Futures Exchange,
2012). In the following section we will present earlier studies on the field. The question
we want to answer follows; will commodity futures increase the risk-‐adjusted return in
efficient portfolios using assets from the Swedish and Norwegian equity markets? Why
we compare two different markets is to open up for analysis and see if our hypotheses
6 Investing in Commodities, 2013
differ between the two areas. We intend to compare the results from the Swedish stock
market with the results from the Norwegian stock market. Our study will be based on
years 2002-‐2012 to receive an “up-‐to-‐date” research within the field, and during a more
concentrated sample period than many of the previous studies mentioned below.
1.1 Earlier empirical work
This section explains the relevance of the thesis contents and shows earlier research
results from different academic sources. The research about combining efficient
portfolio with specific commodities has been done many times before and a majority of
the studies has reached the same conclusions, which is that commodities will provide
the efficient portfolio with a higher average return and a lower average
variance/standard deviation.
Several researchers argue that due to the low correlation between commodities and
stocks/bonds, the diversified portfolio will receive a lower standard deviation
(Georgiev, 2001), or significant return enhancements at all levels of risk (Jensen,
Johnson, & Mercer, 2000). According to previous empirical results; by combining a
diversified portfolio consisting of U.S. stocks and bonds with a commodity index, it
reduced the standard deviation by 0.90 percent while the Sharpe ratio was maintained
(Georgiev, 2001). The study tested the same approach with a global portfolio, then the
standard deviation was reduced by 0.50 percent and the Sharpe ratio did slightly
improve (Georgiev, 2001).
There are several studies that have reached the same results, that the Sharpe ratio will
be higher or maintained with a lower standard deviation. Another study concluded that
an equally weighted portfolio consisting of a commodity index and S&P 500 assets,
received a higher compounded return and lower standard deviation compared to a
stand-‐alone investment of S&P 500 assets, the data that was used stretched from 1969
to 2004 (Erb & Campbell, 2006). The conclusion was that because of the negative
correlation between the assets and low transaction costs the combined portfolio was
more efficient (Erb & Campbell, 2006).
7 Investing in Commodities, 2013
This covariance relationship does often add benefits to the portfolio such as higher risk-‐
adjusted return and lower risk. Including commodity futures to an efficient portfolio
during periods of restrictive monetary policy, enhances return at all levels of risk. While
during periods of expansive monetary policy, including futures to efficient portfolios has
no return enhancements at all (Jensen, Johnson, & Mercer, 2000). However we will not
investigate the effect of monetary policy on commodity investment in this paper.
1.2 Hypotheses
A discussion will be established further on in this paper, we have the intention to
answer some hypothesizes regarding commodity futures, these are essential in order to
find answers and conclusions regarding our general hypothesis; will commodity futures
enhance risk-‐adjusted return in efficient portfolios?
There have been many discussions and speculation regarding gold future investments,
and investors share different point of a view. Earlier empirical work shows that a
combination with stocks/bonds & commodity futures will increase the return-‐to-‐
volatility, since the gold price has increased greatly since the last ten years (Figure A1 in
Appendix), we believe it will be an outstanding investment.
Commodity prices are mainly set by market supply and demand; since the Norwegian
economy is highly based on the oil industry it will probably have high correlation with
the commodity indices. Therefore we believe that the Norwegian portfolio will not
benefit as much as the Swedish portfolio, because of the fact that the correlation
between assets and commodities in Sweden will be generally lower, thereof a higher
return-‐to-‐volatility will be expected.
During the 21st century the global equity markets have suffered from major financial crises.
At the same time emerging economies like China and India started to expand their supply of
physical commodities in order to develop infrastructure etc. This has increased the general
commodity market price and therefore it is possible for speculating investors to make huge
profits. Due to a higher demand of physical commodities and big losses at the global
financial markets, we believe that a stand-‐alone investment in commodities will be superior
8 Investing in Commodities, 2013
in terms of return-‐to-‐volatility compared to other assets like stocks and bonds during our
research period.
The facts and issues presented above are the foundation of the three hypotheses that we
have constructed. As earlier mentioned, in order to create a discussion regarding the main
question, we need a base to proceed from and these hypotheses will accomplish that.
1.2.1 Hypothesis 1
Gold futures will outperform the other commodity futures in terms of return-‐to-‐
volatility.
1.2.2 Hypothesis 2
The Norwegian portfolio will not benefit as much as a Swedish portfolio by adding
commodity futures, in terms of return-‐to-‐volatility.
1.2.3 Hypothesis 3
A stand-‐alone performance of commodity futures during our research period will be better
than stocks and bonds on average.
9 Investing in Commodities, 2013
2 Theory
2.1 Modern Portfolio theory
Modern Portfolio Theory assumes investors to be risk-‐averse i.e. if an investor can
choose between two portfolios with the same return; he will choose the one with lowest
variance or risk. A risk-‐averse investor will avoid adding risky securities to the portfolio,
if not compensated by higher returns depending on the degree of risk-‐aversion
(Investopedia). Before we can construct a portfolio we have to anticipate future returns
of securities and also the variance of the returns. Assuming a risk-‐averse investor, we
will think of future expected return as a wanted thing, and variance of the return as
unwanted. A risk-‐averse investor wishes to maximize the future expected return and to
minimize variance of returns i.e. the risk (Markowitz, 1952).
Further there is an incentive for the investor to diversify among assets and to maximize
the expected return. The investor should therefore diversify funds over the assets
resulting in the highest expected return (Markowitz, 1952, p. 79). The portfolio with the
highest expected return is not necessarily the one with lowest variance, which is what
the investor would try to achieve. The portfolio with highest expected return might also
have a high variance of the expected returns and therefore the investor can reduce
variance by giving up some of the expected return. The E-‐V rule (expected return –
variance) impose that the investor will chose a portfolio that increases the E-‐V
relationship i.e. the portfolio with minimum variance given the expected return, or the
maximum expected return given the variance (Markowitz, 1952, p. 82). In line with the
E-‐V rule, the investor will choose a portfolio somewhere on the efficient frontier and can
maximize the trade-‐off between expected return and variance at some point along this
frontier.
Combining these facts we can conclude that there is a purpose of diversification among
assets which will result in the highest expected return – variance relationship. There are
many ways to measure the return-‐variance relationship, but in this paper we will focus
on the Sharpe-‐Ratio and the mean variance optimization concept to construct portfolios.
10 Investing in Commodities, 2013
2.2 Sharpe Ratio
The Sharpe ratio measures the reward-‐to-‐volatility ratio investing in a risky asset over a
risk-‐free asset. In other words the ratio provides you with the portfolio’s excess return
per unit of risk. It is a risk-‐adjusted measurement that allows you to compare different
assets with different risk and therefore it is a good measurement for portfolio
evaluation. The Sharpe-‐ratio is calculated by dividing the risk-‐premium (the expected
return of the portfolio subtracting the risk-‐free rate) with the portfolios standard
deviation. The Sharpe ratio could get a negative value, but then the asset is
underperforming the risk-‐free rate. (Bodie, Kane, & Marcus, 2011, pp. 161, 234)
𝑆𝑅 =𝑅! − 𝑅!𝜎!
2.3 Portfolio optimization
The goal within modern portfolio theory is to optimally allocate your invested funds
between different assets. The mean-‐variance optimization (MVO) is a quantitative
analysis tool, which takes the risk to volatility measure into account when allocating
resources. The target is to maximize the mean return for the portfolio at the lowest level
of risk, or to minimize the level of risk at a given level of return. Optimization is highly
dependent on the covariance between the assets, so an asset used for hedging purposes
must have negative correlation with the assets itself. If a portfolio has less than perfectly
correlated assets it will always provide better risk-‐return relationship than holding the
individual assets by themselves. As the correlation becomes more negative the greater
the gains in efficiency of the portfolio are. (Bodie, Kane, & Marcus, 2011, p. 232)
2.3.1 Optimal risky portfolio
The optimal risky portfolio is a combination of risky assets that gives the best risk-‐
return trade-‐off (Bodie, Kane, & Marcus, 2011, p. 224). At the point where the Capital
Allocation Line (CAL) tangents the efficient frontier we find our optimal risky portfolio,
and also the highest possible Sharpe-‐ratio of the portfolio.
𝑀𝑎𝑥 (𝑅! − 𝑟!)
𝜎! 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑤! = 1
11 Investing in Commodities, 2013
N symbolizes the number of assets in the portfolio.
2.3.2 Portfolio Variance
The variance of a portfolio consisting of 2 risky assets could be calculated through:
𝜎!! = 𝑤!!𝜎!! + 𝑤!!𝜎!! + 2𝑤!𝑤!𝐶𝑜𝑣(𝑅!𝑅!)
However, as the number of assets increases the number of covariance terms increases
rapidly and the matrix grows with one grade for each asset added. The variance for a
portfolio consisting of n assets could be written as 𝑊𝑣𝑊! where W is the column vector
containing the different weights of the assets, V is the covariance matrix of the assets
and 𝑊! is the transpose of the matrix W (Pareek, 2009).
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝑤!…𝑤! 𝑥𝜎!! ⋯ 𝜎!!⋮ ⋱ ⋮𝜎!! ⋯ 𝜎!!
𝑥𝑤!⋮𝑤!
One can simplify this optimization problem using Excel add-‐ins which we present in
Appendix 11.1.
2.4 Commodities
A commodity is a physical item that is usually used as an input to produce other goods
or services. However, a commodity is as well a traded financial asset on the commodity
exchange markets in the same way as other financial securities e.g. bonds and stocks
(Investopedia, 2012).
The spot prices of commodities are mainly determined by supply and demand. For
instance, during the financial crisis in 2007 when the global economy went into a bad
recession the price on rice went up significantly (The World Bank, 2012). This kind of
scenario could be explained by; during that time period the consumers had less cash-‐
flow and therefore they will consume cheaper products. However, if a lot consumers act
the same way, the demand on rice will go up which will result into that the price will
increase.
12 Investing in Commodities, 2013
2.5 Commodity investment vehicles
2.5.1 Exchange Traded Funds (ETF)
An easy way to get access to the commodity markets is to invest in an Exchange traded
commodity fund. Funds like these may be iShares GSCI Commodity-‐Indexed Trust Fund
(iShares, 2012) or PowerShares DB Commodity Index Tracking Fund (Deutsche Bank,
2012). These funds have the objective to track a commodity index related to each fund.
Buying shares in an exchange traded index tracking fund allows the investor to “buy the
market” within a single investment. Instead of investing in many single commodity
futures, the investor can use an ETF to receive a diversification among many sectors of
commodities.
2.5.2 Mutual Funds
Another was for an investor to get easy access to commodity markets is to invest in a
mutual fund. A mutual fund pools together funds from many investors to invest in assets
such as stocks, bonds and other derivatives e.g. commodity futures (Investopedia, 2012).
2.5.3 Equities in commodity based companies
The most common or traditional way for investors to get access to the benefits of
commodity exposure is to invest in companies which operate in a commodity intense
industry (Jensen & Mercer, 2011, pp. 3-‐4). E.g. investing in Lundin Mining will give the
investor exposure to precious metals prices, or investing in Statoil will give you
exposure to oil prices. It is important to point out that this kind of investment is not only
dependent on commodity prices, but also the performance of the company itself. Even if
oil prices are rising, it does not necessarily mean that the Statoil stock will rise.
13 Investing in Commodities, 2013
2.5.4 Future contract
A future contract is a standardized contract that allows the investor to buy or sell an
asset in the future at a pre-‐determined price i.e. the future price (Hull, 2011). Generally
these transactions are made on the future exchange and the underlying assets are
commonly a commodity or another kind of financial instrument. As mentioned before,
these contract are standardized which implies that certain requirements must have
been established, e.g. the quality of the asset, the amount, the delivery date and location
of the delivery (Hull, 2011).
Speculating investors heavily trade this kind of contracts, however the actual delivery of
the underlying asset rarely happens. Investors tend to close out their position before the
maturity of the contract, in order to make profits (Hull, 2011). The difference between
the actual spot price of the asset, and futures price of the contract is called the basis. At
the expiration date of the contract the basis should be zero if the no arbitrage condition
holds, but before the expiration the basis may be both positive and negative which
exposes the investor to a kind of risk, basis risk (Hull, 2011).
2.6 Goldman Sachs Commodity Index (GSCI)
“The S&P GSCI is a composite index of commodity sector returns representing an
unleveraged, long-‐only investment in commodity futures that is broadly diversified across
the spectrum of commodities” (Goldman Sachs, 2013). The index is divided into five
different commodity-‐types (Energy, Industrial-‐metals, Precious-‐metals, Agriculture and
Livestock) where Energy is the highest weighted with almost 80% of the index
(Morningstar, 2007). The index is world-‐production weighted which means that the
weight in the index of each commodity is determined by the average produced quantity
of each commodity the last five years.
14 Investing in Commodities, 2013
2.7 Test for normally distributed returns
The mean variance optimization assumes the returns to be normally distributed i.e. that
there is no skewness in the distribution of returns. Testing for this assumption it is
possible to use Shapiro-‐Wilk test for normality by first stating a null hypothesis and an
alternative hypothesis.
𝐻! = 𝑇ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 𝑎𝑟𝑒 𝑛𝑜𝑟𝑚𝑎𝑙𝑙𝑦 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑑
𝐻! = 𝑇ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑛𝑜𝑟𝑚𝑎𝑙𝑙𝑦 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑑
If the p-‐value of the test is lower than the chosen significance level (5%) it is not
possible to reject the null hypothesis, i.e. one can conclude that the data is normally
distributed. A test statistic (W) close to one indicated that the data is normally
distributed as well. The test statistic for Shapiro-‐Wilk test is
𝑊 =( 𝑎!𝑥(!))!
!!!!
(𝑥! − 𝑥)!!!!!
𝑥(!) is the ith order statistic (smallest number in the sample)
𝑥 is the sample mean
𝑎! constants are given by 𝑎!,… . ,𝑎! = !!!!!
(!!!!!!!!!)!/!
𝑚 = (𝑚!,… . ,𝑚!)! 𝑚! are the expected values of the order statistics of independent and
identically distributed random variables sampled from the standard normal
distribution.
𝑉 is the covariance matrix of the order statistics
15 Investing in Commodities, 2013
2.8 Criticism of MPT
As with many theories, modern portfolio theory is based on certain assumptions to
make the model applicable in practice. Sometimes assumptions can be fully possible to
achieve in theory but not in the real world. Examples of these assumptions are that stock
returns generally follow a normal distribution, which has been proven not true and
which we later on provide evidence with the same conclusions for (Fama, Jan, 1965).
MPT assumes all investors to be price-‐takers and cannot affect stock prices. In reality it
is possible to affect prices by selling or buying enough amounts of an asset to affect
market prices up or down. Other assumptions such as investors are rational and have
access to the same information have been criticized as well (Elton, Gruber, & Busse,
2004). Further on, critique of the Capital Asset Pricing Model has been presented
showing that mean-‐variance efficiency of the market portfolio and CAPM equations are
equivalent and that any mean-‐variance efficient portfolio will satisfy the CAPM equation.
The market portfolio, which is vital in the CAPM equation, is not possible to achieve; it
would include every possible asset in the world such as stocks, bonds, precious metals,
jewelry or anything with value. This portfolio is not observable and therefore investors
often use market indices as a proxy for this portfolio which leads to a discussion about
the validity of CAPM (Roll, 1977).
Many financial theories are based on assumptions that may be impossible to achieve in
practice, but to be able to use the theories one must simplify observations in the real
world. If we would take every little thing into account when making a model, the model
would not be applicable, since there would be an infinite number of inputs. What we can
do when trying to model real world scenarios is to take into account as much as
possible, without making the model to complicate.
However, one can argue that if the model is not applicable in practice, or that is based on
assumptions which cannot be satisfied, why make the model in the first case? And this is
of course not a problem designated to only modern portfolio theory, but to all models
describing the real world.
16 Investing in Commodities, 2013
3 Data
3.1 Stocks
To construct two standard portfolios for each country we first needed to select our
stocks in the portfolios. We have chosen fifteen major companies operating in different
sectors from each of the stock markets, which should represent a well-‐diversified stock
and bond portfolio (Table A 1,Table A 2). We have used monthly historical price data
during time period 2002-‐2012, which we received from Bloomberg. We have chosen
monthly data because of lack of observations from the daily data. This should not affect
our results, because we calculate an average over the sample period. In addition to our
selection of stocks and corporate bond funds, we have chosen to add commodity future
contracts; gold, oil, coffee, rough rice, silver and cooper to evaluate whether we are able
to increase the value of our portfolios without any additional risk exposure.
3.2 Commodities
Our selection of futures was based on the most heavily traded commodities by today’s
measure. We also considered using commodity futures from different sectors to
investigate if there was a general pattern for all commodities.
Table 1 shows how our chosen commodity future contracts are measured relatively to
their future price (CME Group, 2013), we have not considered the contract size. For
instance, if you wanted to buy a future contract with copper, one COMEX contract
contains the amount of 37,500 pounds. This was essential to further on calculate the
expected return of each commodity.
Table 1
Price AmountGold (GC1) US dollars 100 troy ounces (≈ 3 110 gram)Oil (CO1) US dollars per barrel (≈ 159 liters)Coffee(KC1) US cents per pound (≈ 454 gram)Copper US cents per pound (≈ 454 gram)Silver(SI1) US dollars troy ounce (≈ 31 gram)Rough Rice (RR1) US dollars 100 pounds (≈ 45 kg)
17 Investing in Commodities, 2013
3.3 Corporate Bond Funds
Having some troubles finding data on corporate bonds over our full sample period, we
decided to use two major mutual funds investing in corporate bonds as a proxy. Since
Carnegie corporate bond fund invest mainly in Nordic corporate bonds (Carnegie
Investment Bank AB, 2012), we found it appropriate to use it in our Swedish portfolio
and our Norwegian portfolio. SEB fund 5 – SEB corporate Bond Fund, invests in corporate
bonds in OECD countries, the fund receives its interest rate risk from the Swedish
market (Morningstar, 2013).
3.4 Risk-‐free rate
The Swedish risk-‐free rate was calculated from a 3 month Government Bond (GSGT3M
Index). The Norwegian risk-‐free rate was calculated in the same way, with a 3 month
Government bond (GNGT3M) but it was issued by the Norwegian government instead.
The data was collected through Bloomberg, both of the rates represents the same time
period as the other chosen assets, i.e. 2002-‐2012. We considered using other proxies for
the risk-‐free rate such as STIBOR and NIBOR, which are the Inter Bank Offered Rates
within each country. However, we ended up with more appropriate values using the 3
month bond; the STIBOR rate was surprisingly high in our opinion.
Nevertheless, the risk-‐free rate in Norway exceeded the Swedish by almost 0.7% at an
annually basis. This will of course affect our results in terms of the expected return for
each created portfolio, because the general risk premium in Norway will be lower.
In a previous study the authors examined older data from a larger sample period (1973-‐
1997) (Jensen, Johnson, & Mercer, 2000). Our approach is to study more up-‐to-‐date data
over a shorter sample period since commodity trading has grown massively the last 15
years.
18 Investing in Commodities, 2013
4 Methodology
To begin with we wanted to evaluate how important the role of commodity futures was
in a well-‐diversified Swedish stock and bond portfolio, over different levels of risk. By
important we mean higher risk-‐adjusted return in terms of the Sharpe ratio. To get a
comparable result we added a Norwegian portfolio consisting of stocks and bonds which
allowed us to gather certain differences and similarities between the two markets.
We began by computing an index with 2002 as starting point to see graphically the
correlation between the Swedish and Norwegian stock markets, and the GSCI Index. In
Figure 2 we used the GSCI Index as a proxy for all our commodity futures since it would
get a better overview of the correlation between the two Nordic markets and the
commodity returns. The graph shows clearly that the OSEBX index is more correlated
with the GSCI Index, and the OMXS30 is somehow correlated with GSCI Index but only
weakly.
4.1 Bloomberg add-‐ins
As mentioned earlier our data comes from Bloomberg and was added in Excel using the
Bloomberg add-‐in. To collect the monthly stock prices we used “Historical end of day
wizard” and specified which asset we wanted data from. We used the “Last Price”
(PX_LAST) which for equities is the last price provided by exchange and for futures the
last price traded until the settlement price is received.
4.2 Optimal Portfolio Allocation
The focus in our methodology part is mainly based on portfolio optimization, where we
intend to maximize the Sharpe-‐ratio. To be able to use our data we had to first calculate
the monthly returns of each asset from its price and also the standard deviation and
average return which is used in the optimization part. From the monthly return data we
used the Data analysis tool in Excel to build a covariance matrix for each of our
portfolios including the new commodity.
19 Investing in Commodities, 2013
In order to construct our efficient portfolios based on modern portfolio theory with
maximized risk-‐adjusted return, we used the formulas presented in the theory part, but
had Excel making the calculations since the portfolio variance would contain a huge
number of terms. This calculation is a trivial task when only using two assets, but as the
number of assets increases, the number of covariance terms increases rapidly. To
simplify these calculations we used Excel VBA code and also the Solver function. The
target was to maximize the Sharpe-‐ratio by changing the weights for the assets in our
portfolio, under the constraint that the sum of the weights could not be larger than
100%, e.g. we don’t allow leveraged positions.
We received the optimal weights among our chosen assets, but we wanted to see how
the weights changed over different levels of risk. Initially we thought about creating the
optimal risky portfolio and the minimum variance frontier for each of the
stock/commodity portfolios. However we received unexpectedly low values for the
monthly standard deviation and therefore we decided to look further on how the
expected return and Sharpe-‐ratio evolved over different levels of risk. Still maximizing
the Sharpe ratio but adding constraints to the solver that the cell containing standard
deviation should be equal to certain values ranging from 0,5% to 5% of monthly
standard deviation. This procedure was repeated for each combination of
commodity/portfolio.
4.3 Test for normally distributed returns
In order to investigate whether the returns of each asset class were normally distributed
we computed the Shapiro-‐Wilk test for normality using Stata.
20 Investing in Commodities, 2013
5 Results
In this part we will summarize our results from the optimal allocation of assets from the
methodology section. The results were almost as we expected in theory. We did not
allow for short-‐selling when maximizing the Sharpe Ratio subject to the constraints,
which caused some asset weights to be equal to zero. Since we did not allow for short-‐
selling we did neither receive any leveraged positions among our chosen assets because
of the sum of the weights in our portfolio should be equal top 100%. Therefore our
results will be based on no short-‐sell positions or leverage positions.
Figure 1
This evidence does support Hypothesis 2 that the Norwegian portfolio would not benefit
as much from adding commodity futures as the Swedish one.
Table 2
The correlation matrix also provides evidence that the Norwegian stock market is more
correlated with the commodity index GSCI.
SPGSCI Index OMX Index OSEBX IndexSPGSCI Index 1,00 OMX Index 0,17 1,00 OSEBX Index 0,46 0,77 1,00
21 Investing in Commodities, 2013
5.1 Swedish results
First we will present the results based on the Swedish portfolio.
5.1.1 Asset Allocation
In Table 3 we present the asset allocation of the Swedish portfolio consisting of stocks, government bonds and corporate bonds (Henceforth referred to as Swedish standard portfolio).
Table 3
Table 4 displays the asset allocation of the Swedish standard portfolio combined with
commodity futures (Henceforth referred to as Swedish basket portfolio) Table 4
Swedish standard portfolioPortfolio std dev Stocks Govt. Bonds (3m) Corporate bond
0,25% 3,23% 64,04% 32,73%0,50% 6,25% 30,43% 63,32%1,00% 14,69% 0,00% 85,31%1,50% 23,82% 0,00% 76,18%2,00% 32,41% 0,00% 67,59%2,50% 40,82% 0,00% 59,18%3,00% 49,13% 0,00% 50,87%3,50% 57,39% 0,00% 42,61%4,00% 65,62% 0,00% 34,38%4,50% 73,83% 0,00% 26,17%5,00% 82,02% 0,00% 17,98%
Swedish portfolio with commodity futuresPortfolio std dev Commodity Futures Stocks Govt. Bonds (3m) Corporate bond
0,25% 2,30% 3,13% 64,62% 29,94%0,50% 4,74% 5,84% 32,21% 57,21%1,00% 10,58% 12,59% 0,00% 76,83%1,50% 16,63% 19,74% 0,00% 63,63%2,00% 22,44% 26,85% 0,00% 50,71%2,50% 28,21% 33,82% 0,00% 37,98%3,00% 33,92% 40,70% 0,00% 25,37%3,50% 39,61% 47,55% 0,00% 12,84%4,00% 45,27% 54,37% 0,00% 0,36%4,50% 45,70% 54,30% 0,00% 0,00%5,00% 46,25% 53,75% 0,00% 0,00%
22 Investing in Commodities, 2013
Figure 3 is a comparison of the performance of the two Swedish portfolios over different
levels of risk. We can see that the Swedish basket portfolio generates higher return at
every given level of risk than the Swedish Standard portfolio.
Figure 2
Figure 3
0,00%
0,50%
1,00%
1,50%
2,00%
2,50%
0,00% 1,00% 2,00% 3,00% 4,00% 5,00% 6,00%
Monthly expected return (%
)
Monthly standard deviation (%)
Swedish Standard Portfolio vs. Basket Portfolio
Standard Portfolio
Basket Portfolio
0,00%
0,40%
0,80%
1,20%
1,60%
2,00%
2,40%
0,25% 0,50% 1,00% 1,50% 2,00% 2,50% 3,00% 3,50% 4,00% 4,50% 5,00%
Monthly expected renturn (%
)
Monthly standard deviation (%)
Swedish portfolios
Standard
Gold
Oil
Rice
Coffe
Copper
Silver
Basket
23 Investing in Commodities, 2013
Figure 4 was created as evidence to show that by adding a single commodity future to
your portfolio, you will receive a higher return per unit of standard deviation. However,
as expected, a low magnitude of the standard deviation will result in insignificant
differences compared to the standard portfolio. By observing the Figure 3, it is clear that
by adding gold futures, the return remarkably exceed the other chosen futures. The
Standard portfolio including gold futures performs almost as well as the Basket
portfolio.
5.2 Norwegian results
In this section the results from the Norwegian portfolio allocation will be presented.
5.2.1 Asset allocation
Table 5 displays an overview of the asset allocation of the Norwegian portfolio consisting of Norwegian stocks, government bonds and Nordic Corporate bonds (Henceforth referred to as Norwegian standard portfolio)
Table 5
Table 6 displays the asset allocation of the Norwegian portfolio consisting of Norwegian
stocks, government bonds, Nordic corporate bonds and commodity futures (Henceforth
referred to as Norwegian basket portfolio).
Norwegian standard portfolioPortfolio std dev Stocks Govt. Bonds (3m) Corporate bond
0,25% 2,63% 66,64% 30,73%0,50% 4,77% 33,11% 62,12%1,00% 10,87% 0,00% 89,13%1,50% 16,92% 0,00% 83,08%2,00% 22,62% 0,00% 77,38%2,50% 28,20% 0,00% 71,80%3,00% 33,90% 0,00% 66,10%3,50% 39,57% 0,00% 60,43%4,00% 44,85% 0,00% 55,15%4,50% 50,48% 0,00% 49,52%5,00% 56,10% 0,00% 43,90%
24 Investing in Commodities, 2013
In Table 6 we received some unexpected results that the weight of commodity futures in
the Norwegian portfolio became generally high. We expected the weight of futures in the
Norwegian portfolio to be less than the Swedish portfolio since the Norwegian stock
index has higher correlation with the GSCI index. Table 6
The performance of the Norwegian Basket portfolio does clearly outperform the
Norwegian Standard portfolio. As the level of risk increases, the larger the difference
between the Standard portfolio and the Basket portfolio gets and the more weights in
futures are requested well.
Figure 4
Norwegian portfolio with commodity futuresPortfolio std dev Commodity Futures Stocks Govt. Bonds (3m) Corporate bond
0,25% 3,56% 1,04% 72,00% 23,39%0,50% 7,64% 1,73% 40,24% 50,40%1,00% 17,44% 2,78% 0,00% 79,79%1,50% 28,07% 4,67% 0,00% 67,27%2,00% 37,88% 6,72% 0,00% 55,39%2,50% 47,57% 8,70% 0,00% 43,73%3,00% 57,16% 10,66% 0,00% 32,18%3,50% 66,69% 12,58% 0,00% 20,73%4,00% 76,21% 14,52% 0,00% 9,27%4,50% 82,75% 17,25% 0,00% 0,00%5,00% 79,65% 20,35% 0,00% 0,00%
0,00% 0,20% 0,40% 0,60% 0,80% 1,00% 1,20% 1,40% 1,60% 1,80%
0,00% 1,00% 2,00% 3,00% 4,00% 5,00% 6,00% Monthly expected return (%
)
Monthly Standard Deviation (%)
Norwegian Standard Portfolio vs. Basket Portfolio
Norwegian Portfolio
Basket Portfolio
25 Investing in Commodities, 2013
At a 3% monthly standard deviation the standard portfolio will have 0,851% in monthly
return, and the basket portfolio will have a return of 1,224%. This means that the Basket
portfolio would have 44% higher return than the Standard portfolio consisting of
Norwegian stocks and Nordic corporate bonds. We have also concluded that the Sharpe
ratio is higher for the Basket portfolio at each level of standard deviation.
Figure 5
One can obviously see that at any given level of risk the Basket portfolio outperforms all
other combinations of assets. At a 5 % level of monthly standard deviation, gold futures
requests almost 40% of the allocation of the basket portfolio which was not very
surprisingly. What surprised us though is that Brent Oil futures allocate 15% of our
optimal basket portfolio, since Oil futures have high correlation with many Norwegian
companies producing oil products. On the other hand, when maximizing the allocation of
Norwegian stocks, we did not receive any weights in oil based companies.
0,00%
0,40%
0,80%
1,20%
1,60%
2,00%
0,25% 0,50% 1,00% 1,50% 2,00% 2,50% 3,00% 3,50% 4,00% 4,50% 5,00%
Monthly expected return(%
)
Monthly standard deviation(%)
Norwegian portfolios
Standard
Gold
Brent Oil
Rice
Coffee
Copper
Silver
Basket
26 Investing in Commodities, 2013
6 Analysis
Because of the fluctuations within the global economy, investors seek different
alternatives to reduce their exposure to risk. Since the financial crisis in 2007, many
precautions were made in order to withstand another crisis. There has been a debate
regarding speculators being a key reason for the financial crisis 2007. A result of this
was the Volcker Rule which restricted US commercial banks from speculating in certain
risky investments, such as commodity derivatives, using deposits to trade on its own
account (Financial Times, 2012). However, from a portfolio diversification point-‐of-‐
view, commodity futures can be very useful as an investment instrument.
As mentioned earlier in this paper, in general, commodity futures are quite uncorrelated
with the market i.e. as Table 2 indicates the correlation between SPGSCI and OMX30 is
0.17. Why does this correlation generally differ from other more traditional asset
classes? Because of the fact that commodity prices are mainly set up by global supply
and demand and equity markets are highly dependent on the performance of specific
industries and sectors. For instance, if an economic recession occurs within a country it
is likely that an equity index e.g. OMX30, will perform poorly but this does not
necessarily mean that commodity prices will decrease. Hence, we can state that the
correlation between commodities and equities is low. We have concluded within this
paper that by adding commodity futures to a stock and bond portfolio the investor will
receive a higher risk-‐adjusted return, at least within out research period and our chosen
geographical areas.
Still, a lot of previous studies have tried to evaluate the historical performance of
commodity futures as a stand-‐alone investment with different results and conclusions.
In a study based on 1973-‐1997 data, the performance of commodity futures as a stand-‐
alone investment is concluded to be inferior to other asset classes (Jensen, Johnson, &
Mercer, 2000). However, in a new updated study based on 1970-‐2009 data , evidence is
provided that the stand-‐alone performance of commodity futures has higher returns but
higher standard deviations as well, which resulted in a risk-‐return relationship in line
with the equity markets (Jensen & Mercer, 2011, pp. 6-‐11).
27 Investing in Commodities, 2013
In our results we have established that the commodity futures have in general been a
superior investment to the equity markets during the 21st century. The general return of
commodity futures has been higher than the equities but the standard deviations have
also been higher, basically we received the same results as previous studies (Jensen &
Mercer, 2011). On a risk-‐adjusted return basis the commodity futures during our
research period have been superior to the equities with a few exceptions.
Since our study is based on the latest ten years (2002-‐2012), it is not unexpected that
we achieve these results. Prior to year 2000, the commodity prices have been relatively
low (Center for Futures Education, Inc., 2013), but since increasing demand in emerging
economies like China and India, the commodity prices have risen substantially. In a
previous study (Jensen & Mercer, 2011) the authors used a larger sample period
including both less and more volatile years of commodity prices, which explains why
their results indicates that commodity futures have been a relatively good stand-‐alone
investment.
In a previous study they used a different time period than this paper, but our data is
even more updated and it is concentrated on the Scandinavian markets instead of
American markets (Jensen & Mercer 2011). However, there is some kind of a pattern
that commodity futures in general as a stand-‐alone investment the latest 10 years has
been a relatively good investment compared to stocks. Though, commodity futures are
more commonly used as a portfolio component rather than a stand-‐alone investment,
because it is a good option for investors who want to diversify their portfolio in order to
reduce market risk.
Overseeing the results in the previous section, we can clearly state for both countries
that by adding several commodity futures to the standard portfolio, the return-‐to-‐
volatility relationship will increase. It was expected that both portfolios would benefit
from this additional investment, in terms of lower risk and higher return. However, it
was interesting that the Norwegian portfolio requested more weights in commodity
futures when optimizing the portfolio allocation than the Swedish portfolio. Initially we
thought that the Swedish portfolio would have a higher proportion of futures than the
Norwegian, because of the relatively low correlation between OMX30 and the GSCI,
28 Investing in Commodities, 2013
which is stated in Hypothesis 2. What we did not consider was that the stocks in the
Swedish portfolio were a good investment opportunity compared to the Norwegian
stocks, in terms of return-‐to-‐volatility.
Observing Table 3 and 5, we can point out as the standard deviation increases; the
Swedish portfolio requests significantly more weights in stocks compared to the
Norwegian portfolio at all specified levels of risk. When we stated the hypotheses in the
beginning of the paper, we did not think about the individual performance of each stock,
but selected the stocks from each equity market representing a well-‐diversified
portfolio. In general the Swedish stocks performed far better in terms of return-‐to-‐
volatility compared to the Norwegian stocks. Therefore, by comparing Table 4 and 6 we
can clearly see that the Swedish portfolio requested less percentage of commodity
futures. Though, both of the Swedish portfolios had a higher Sharpe ratio than the
Norwegian portfolios on every level of risk. This could be explained by; during this
specific time period the Swedish stocks performed better compared to the Norwegian
stocks. A reason for this might be that during the financial crisis in 2007, oil spot price
decreased by 68 % June – December (The World Bank, 2013) which affected the
Norwegian economy and equity market more than the Swedish equity market. This can
be observed from Figure 1 that during 2008 the GSCI (which contains almost 80%
weight in oil) fell by 55 %, and the OSEBX index fell by 47 %.
It is also important to acknowledge that the Norwegian risk-‐free rate was almost 0.1%
higher on a monthly basis (1.2 % annually) i.e. this results in a lower risk premium,
which reduces the Sharpe ratio. To state the importance of this matter we used the
Norwegian risk-‐free rate when calculating the Swedish basket portfolio to see what
would happen with the Sharpe-‐Ratio. If both countries would have the same risk-‐free
rate our results would be the other way around, that the Norwegian portfolios would
have higher Sharpe-‐ratios than the Swedish portfolios. This calculation was made only
in illustrating purpose to show the significance of the risk-‐free rate and its effect on the
Sharpe-‐Ratio.
From figure 3 and 5 it is obvious that the Norwegian portfolio benefits more than the
Swedish portfolio in terms of increased return. For instance, by adding commodity
29 Investing in Commodities, 2013
futures to the Norwegian standard portfolio at a 3% level of risk, as an investor you will
receive almost 44% higher return. During the same circumstances concerning the
Swedish portfolio, the investor will only get 12% higher return.
The conclusions about Hypothesis 1 regarding gold were correct. We observed from the
results that gold took a major weight among the futures in both of our portfolios when
maximizing the risk-‐adjusted return. On a 5% monthly standard deviation gold stood for
40% of the weight in the Norwegian Basket portfolio. When comparing the Basket
portfolio with the Standard portfolio including gold futures, we can observe Figure 6 and
state that the risk-‐return tradeoff is very similar. So why not use only gold to increase
the return of your portfolio? One might argue that gold will provide a good
diversification to a portfolio consisting of stocks and bonds by overseeing this historical
data, but for diversification purposes this kind of strategy would be far from the best
option. Investors seeking alternative investments like commodity futures in order to
diversify their portfolios should not limit themselves to one single asset. By invest in a
broader spectrum of commodities or in a commodity index, the investor will receive a
better diversification within the portfolio. We have concluded that there is a slightly
better risk-‐return reward by investing in a basket of commodity futures rather than by
using only gold futures in the portfolio.
Through this paper it is important to clarify that we have excluded a lot of reality factors
concerning commodity trading. We have not included factors like: transaction cost,
commission, broker fees or taxes. Because the access to this kind of information is
limited and including those factors in the calculations would have been very difficult. It
is also necessary to mention that we have no leverage or shortage position in any single
case. If considering these costs for trading single commodity futures, a private investor
with a limited amount of funds would not benefit as much from the investment because
of a major part of the return will be reduced by fees. A better and cheaper way for a
private investor to receive diversification among many commodities would be to invest
in an index tracking ETF and pay a small percentage annual fee.
30 Investing in Commodities, 2013
7 Conclusions
As mentioned earlier through the paper, adding commodity future contracts to the
portfolio is a relatively new phenomenon and since the last 15 years the trading
volumes has exploded at the future markets. In 2011, the commodity futures trading
volume at CBOE Futures Exchange increased by 174 % (CBOE Futures Exchange, 2012)
compared to 2010, corresponding to several millions of future contracts.
By analyzing our results we can conclude that two out of the three hypotheses we stated
were correct. A few of our assumptions and predictions were accurate, but as stated we
did not consider the individual performance of each asset and how it would affect the
proportion of commodities within the portfolio.
We provide certain evidence that the stand-‐alone performance of commodities have
been superior to equities during our sample period, and we believe this is mainly due to
increasing demand from emerging economies and an unchanged supply in the world as
a whole. By our measurements and sample period we can conclude that gold has been a
superior commodity investment during the last decade in terms of higher return and
lower risk compared to the others. However, it is not efficient “to put all of your eggs in
the same basket”, a combination of commodity futures are therefore the optimal choice
in terms of reducing the overall exposure to risk.
It is not likely for an investor in the real world to invest 80% of their funds in
commodities and the rest in equities. One of the purposes of this paper was to display
what happens with the allocation of each asset over different levels of risk. When
constructing our optimal risky portfolios we received an allocation of 2.7% of future
contracts in the Swedish Basket portfolio and 12.1% in the Norwegian basket portfolio.
This might represent a more traditional allocation of commodities in portfolios as many
studies has concluded before, that the optimal allocation of commodity futures in
efficient portfolios usually vary between 5 to 10% (Du, 2005, pp. 192-‐194)
Comparing two countries within the same geographical area was essential in order to
see if we could find a pattern regarding the benefits of adding commodity futures to a
31 Investing in Commodities, 2013
diversified portfolio. We concluded that there is a pattern for both countries in terms of
that the return-‐to-‐volatility increased, but the Swedish Basket portfolio got a higher
Sharpe ratio than the Norwegian portfolio at every specified level of risk. On the other
hand, overall the Norwegian equities performed poorly in comparison to the Swedish
which might explain why the Norwegian Basket portfolio surprisingly requested a lot
more allocation of the commodities.
Whether commodity futures will provide high returns in the upcoming years or not is
highly uncertain and it is a subject which is analyzed on a daily basis. Though,
commodities will probably continue to provide benefits of diversification to equity
portfolios in the future.
32 Investing in Commodities, 2013
8 Methodology critique
In this paper we are aware of that some of our data and calculations might not reflect a
real world scenario. We are aware of that the corporate bond funds that we have used
might not be an optimal choice as a proxy to represent actual corporate bonds, but this
was a last solution in order to receive a diversification beyond only stocks. Through the
statistic Shapiro-‐Wilks test, we have noticed that the return data we collected from
Bloomberg.com was not normally distributed. Since, mean-‐variance-‐optimization
assumes that the returns of the assets are normally distributed; this may cause biasness
within our results. However, usually the stock returns are not normally distributed, in
other terms there is some skewness of the distribution among the returns (Ford, 2012).
As we mentioned earlier within this paper, we have excluded a lot of reality factors
regarding commodity trading; transactions cost, commissions and taxes. If these factors
were included, we would get different results, especially regarding the return of each
asset.
9 Further research
There are a few excluded factors we could have used in order to get more reliable
results. A good extension of this paper would be if we included factors like transaction
costs, taxes and commissions. It would also be interesting to investigate how the
monetary policy could affect the allocation and efficiency of commodity futures, as our
inspiring paper did (Jensen & Mercer, 2011). Then we could see if we got any similar
results as they did, and compare both papers if there were certain differences, that
would be a great addition to the analysis. It would have also been interesting to see what
would happen with the allocation of the assets if we added other securities like; real
estate’s investment trusts and currencies. We are highly confident that the results would
differ a lot and that the allocation would be completely reallocated.
33 Investing in Commodities, 2013
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37 Investing in Commodities, 2013
11 Appendix
In the appendix we will present tables and figures which are not suited in the thesis, but still provide necessary information regarding the results.
Figure A1 is a graphical illustration, where Gold Futures and the Goldman Sachs Commodity Index have been indexed with 2002-‐01-‐01 as starting value. From the figure we can see that gold futures have outperformed GSCI, which we use as a comparison for the rest of our commodities. This supports Hypothesis 1.
Figure A 1
0
1
2
3
4
5
6
7
1-‐1-‐2002
2-‐1-‐2003
3-‐1-‐2004
4-‐1-‐2005
5-‐1-‐2006
6-‐1-‐2007
7-‐1-‐2008
8-‐1-‐2009
9-‐1-‐2010
10-‐1-‐2011
Gold Futures
GSCI
38 Investing in Commodities, 2013
Table A 1
Table A1 displays our chosen Swedish company stocks and which sector they operate within.
Table A 2
Table A2 displays our chosen Norwegian company stocks and which sector they operate within.
HMB (Hennes & Mauritz AB) apparel retail
VOLVB (Volvo AB)construction and farm machinery; heavy trucks
ASSAB (Assa Abloy AB) building productsTEL2B (Tele2 Ab) telecomSEBA (SEB AB) commercial bankMTGB (Modern Times Group AB) broadcasingINVEB (Investor AB) multi-‐sector holdingsSWMA (Swedish Match AB) tobaccoTLSN (Telia Sonera AB) telecomAZN (Astra Zeneca AB) pharmaceuticalsERICB (Ericson AB) telecomNDA (Nordea Bank AB) commercial bankSCAB (Svenska Cellulosa AB) paper productsBOL (Boliden AB) diversified metals and miningLUPE (Lundin Petroleum AB) petroleum
STL (Statoil ASA) petroleumNHY (Norsk Hydro ASA) petroleumORK (Orkla ASA) multi-‐sector holdingsTEL (Telenor ASA) telecomATEA (Atea ASA) ITNSG (Norske Skogsindustrier ASA) paper productsDNB (Den Norske Bank ASA) commercial bankGOD (Goodtech ASA) electric utilitySCI (Scana Industrier ASA) industrialTOM (Tomra ASA) recyclingSUBC (Subsea7 ASA) petroleumEVRY (EVRY ASA) information technologyBON (Bonheur ASA) holding companyRCL (Royal Caribbean Cruisses ASA) hospitality, tourismTGS (TGS Nopec Geophysical ASA) geoscience data
39 Investing in Commodities, 2013
11.1 Excel and VBA
To simplify our optimization problems we used Excel and Visual Basics for Microsoft
Applications, which is a programming language. The purpose for using VBA in this paper
is to reduce the vast amount of calculations which would have been necessary to
compute the portfolio variance from a covariance-‐matrix of 17 assets. To receive the
portfolio variance it is possible to use the VBA-‐Code (People.Brunel, 2012).
= ((𝑀𝑀𝑈𝐿𝑇 𝑀𝑀𝑈𝐿𝑇 𝑊𝐸𝐼𝐺𝐻𝑇𝑆;𝐶𝑂𝑉𝐴𝑅𝐼𝐴𝑁𝐶𝐸𝑀𝐴𝑇𝑅𝐼𝑋 ;𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸 𝑊𝐸𝐼𝐺𝐻𝑇𝑆 )
This VBA-‐Code will multiply the column vector with the covariance matrix multiplied by
the transpose of the column vector. To receive the portfolio return we used the
= 𝑀𝑀𝑈𝐿𝑇 𝐴𝑉𝐸𝑅𝐴𝐺𝐸 𝑅𝐸𝑇𝑈𝑅𝑁;𝑇𝑅𝐴𝑁𝑆𝑃𝑂𝑆𝐸 𝑊𝐸𝐼𝐺𝐻𝑇𝑆
Weights are in this case the chosen weights from the optimal allocation and Average
return is the average return of each asset over a certain period.
40 Investing in Commodities, 2013
11.2 Test statistics for Shapiro-‐Wilk test for normality
In this part of the appendix we will summarize our tests from Stata and SPSS
Figure A 2 is a histogram from Stata over the distribution of stock returns, which are not
exactly normally distributed. To get an exact result of the normality test we have to
observe the p-‐values for the Shapiro-‐Wilk test.
Figure A 2
Figure A 3 is a Q-‐Q plot of the stock returns from SPSS
Figure A 3
41 Investing in Commodities, 2013
Figure A 4 is a histogram from Stata over the corporate bond returns distribution. The
histogram shows that returns could be taken as normally distributed, however to conclude
whether this statement or not is true we have to rely on the p-values.
Figure A 4
Figure A 5 is a Q-‐Q plot of the corporate bonds return from SPSS
Figure A 5
42 Investing in Commodities, 2013
Figure A 6 is a histogram from Stata displaying the commodity futures returns distribution.
From the histogram we cannot conclude that the returns are normally distributed and again,
we need to rely on the p-value
Figure A 6
Figure A 7 is a Q-‐Q plot of the commodity futures returns from SPSS
Figure A 7
43 Investing in Commodities, 2013
Table A 3 is a summary of statistics used in the Shapiro-Wilk test for normality
Table A 3
Table A 4 contains the test statistics from the Shapiro-Wilk test. Observing the table we can
see that the test statistic W is close to one which could explain that the returns are normally
distributed. However to be sure about this statement, we need to observe the p-values which
in this case are lower than the chosen level of significance (5%) i.e. we can reject the null
hypothesis that the returns are normally distributed.
Table A 4
Summary statistics Variable Obs Mean Std. Dev. Min MaxStocks 129 0.0094631 0.0598137 (-‐)0.1821368 0.190926Coroporate Bonds 129 0.0036416 0.0088096 (-‐)0.0296038 0.0293444Commodity Futures 129 0.0165823 0.0549259 (-‐)0.2371066 0.1469634
Shapiro-‐Wilk W test for normal dataVariable Obs W V Sig (Prob>z)Stocks 129 0.96173 3.916 0.00107Coroporate Bonds 129 0.96033 4.058 0.00082Commodity Futures 129 0.95046 5.650 0.00013